How do I get my iOS app to run code when something specific in a message in Messages happens?

You can setup Personal Automation using Shortcuts on your phone.

Here's a guide:

Shortcuts App > Create Personal Automation > Message

Here you can select the sender you wish to customise this automation for. You can also specify the message you want to receive from this sender to allow the automation to work.

You can play around with it and test it with your own number.

Hope you found this helpful.

Yeah. That works. I'm hoping to be able to call attention to the message in a more obvious way to the person receiving the message. He could put the sender in the contact and set emergency override on.